We all know that the $L^2$ norm
$$ ||\psi||_2 = \sqrt{\sum_i |c_i|^2} $$
of a quantum state $|\psi\rangle = \sum_i c_i |i\rangle$ is always equal to $1$. It is possible to compute the $L^1$ norm
$$ ||\psi||_1 = \sum_i |c_i| $$
by brute force by simply executing the circuit and then summing the square roots of the obtain probabilities associated with each element of the computational basis.
I was wondering if there is some faster way to do that, maybe using an ancilla.
Extra: is there any way to estimate $\sum_i c_i$?
